Volume densities with the mean value property for harmonic functions. (English) Zbl 0818.31003

Let \(U\) be a bounded domain in \(\mathbb{R}^ d\) \((d\geq 2)\) such that \(0\in U\), and let \(\lambda\) denote Lebesgue measure on \(U\). It is shown that there exists a (strictly) positive \(C^ \infty\) function \(w\) on \(U\) such that \(h(0)= \int hw d\lambda\) for every \(h\) in \({\mathcal H}_ b (U)\), the set of bounded harmonic functions on \(U\). Further, if \(\partial U\) is of class \(C^{1+ \varepsilon}\), then it can be arranged that \(\inf w(U)>0\). The smoothness condition on \(\partial U\) cannot be omitted: the authors construct a domain \(U\) such that if \(w\) is a nonnegative measurable function on \(U\) which satisfies \(h(0)= \int hw d\lambda\) for every \(h\) in \({\mathcal H}_ b (U)\), then \(\inf w(U)= 0\). This answers a question posed by A. Cornea at the International Conference on Potential Theory in Nagoya in 1990.


31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
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